3.2 Decomposition method
3.2.1 Decomposing the decomposition effect
This section introduces through an example the methodology subsequently used, and draws heavily on Rothe (2015).
In the remainder of this article, we will focus on the evolution of certain characteristics of the distribution of the quantities of macronutrients consu-
med in Vietnam: average values and quantiles, between 2004 and 2014. Let us concentrate, below, on the number of calories obtained from the consump- tion of carbohydrates per day and per individual. The same reasoning will apply to the number of calories obtained from the consumption of protein or fat. For any household i in year 2004 and any household h in year 2014, we observe an outcome variable: the per capita and per day amount of calo- ries obtained from the consumption of carbohydrates, denoted by Y2004
i and
Yh2014, respectively. These observations are the realizations of two random
variables, denoted by Y2004 and Y2014, whose marginal cumulative distribu-
tion functions, or CDFs, are F2004
Y and FY2014, respectively. Our object of in-
terest is a distribution feature, denoted by ν(F ), where ν(.) is a function from the space of all one-dimensional distribution functions to the real line. The main features we are interested in include the mean, i.e. ν : F →R ydF (y),
and the α–quantiles, i.e. ν : F → F−1(α) = inf {t : F (t) ≥ α} for a given
value of α ∈ [0, 1].
Suppose, for ease of presentation, that we have observed two covaria- tes for each individual in the sample of a given year: for example, food expenditures and location in either urban or rural areas. Of course, the pre- sentation given below can be easily generalized to more than two covariates. We denote the vectors of the two covariates by X2004 = (X2004
1 , X22004) and
X2014= (X2014
1 , X22014), and their joint CDFs by FX2004 and FX2014, respecti-
vely. The decomposition method aims at understanding how the observed difference between the distribution feature ν(F2014
Y ) and ν(FY2004), i.e.
∆ν
Y = ν(FY2014) − ν(FY2004) (3.1)
is related to differences between the distributions F2004
X and FX2014. For
this, we can define the counterfactual outcome distribution FY2004|2014 that combines the conditional distribution in year 2004 with the distribution of covariates in year 2014, as
FY2004|2014(y) =
Z
FY2004|X(y, x) dFX2014(x) (3.2)
where F2004
Y|X(y, x) denotes the conditional distribution of outcome given
values of the covariates in year 2004. In our example, we can interpret
FY2004|2014(y) as the distribution of per day and per capita carbohydrates
consumption after a counterfactual experiment in which the joint distri- bution of the two covariates is changed from year 2004 to year 2014, but the conditional distribution of per day and per capita carbohydrates con- sumption given these characteristics remains that of 2004. One can then decompose the observed between-year difference ∆ν
Y into ∆ν Y = νFY2014 − νFY2004|2014+νFY2004|2014− νFY2004 = ∆ν S+ ∆νX (3.3)
where ∆ν
S is a structure effect, solely due to differences in the conditional
distribution of the outcome given values of covariates between the two years, and ∆ν
X is a composition effect, solely due to differences in the distribution
of the covariates between the two years.
The different elements of the decomposition (3.3) can be easily estima- ted using nonparametric estimates of CDFs. One such strategy, focusing on densities instead of CDFs, is applied in DiNardo et al. (1996) or Leibbrandt et al. (2010). But the application of such a strategy soon encounters the problem of the curse of dimensionality. For a fixed sample size, the precision of the nonparametric estimators deteriorates very rapidly when the number of covariates increases, even if these estimators are free from any specifi- cation error (Silverman, 1986). In addition, it is also interesting to break down the composition effect for the different covariates. This can be easily done using the Oaxaca (1973) and Blinder (1973) approach when focusing on the between-year difference of average outcomes. But the possibility of disentangling the impact of each of the covariates in the composition effect rests on the very restrictive assumption that the data are generated from a linear model. As pointed out by Rothe (2015), in the general case, it is difficult to express the composition effect as a sum of terms which depend on the marginal distribution of a single covariate only. Instead, an explicit decomposition of the composition effect in terms of the respective marginal covariate distributions typically contains “interaction terms” resulting from the interplay of two or more covariates, and also “dependence terms” re- sulting from between-year difference in the dependence pattern among the covariates.
Rothe (2015) proposes to use results from copula theory in order to disen- tangle the covariates’ marginal distributions from the dependence structure among them. Indeed, the CDF of Xt can always be written as
FXt(x) = Ct(FX1t (x1), FX2t (x2)) for t ∈ {2004, 2014} (3.4)
following Sklar’s Theorem (Sklar, 1959). Ct(.) is a copula function, i.e., a
bivariate CDF with standard uniformly distributed marginals, and Ft Xj(.) is
the marginal distribution of the jth component of Xt(Trivedi and Zimmer,
2007). The copula describes the joint distribution of individuals’ ranks in the two components of Xt. The copula accounts for the dependence between the
covariates in a way that is separate from and independent of their marginal specifications. This result holds for continuous covariates. When some of them are discrete, some identifiability issues may arise, that can be solved by making parametric restrictions on the functional form of the copula.
In this context, the decomposition given by Eq. (3.3) can then be gene- ralized as follows. Let k denote an element of the 2-dimensional product set {2004, 2014}2, i.e. k = (k
1, k2) where k1 (resp. k2) is equal to either 2004
setting where the conditional distribution is as in year t, the covariate dis- tribution has the copula function of year s, and the marginal distribution of the lth covariate is equal to that in group k by
FYt|s,k=Z FYt|X(y, x)dFs,k X (x) (3.5) with FXs,k(x) = Cs(Fk1 X1(x1), F k2 X2(x2)). (3.6)
For instance, the counterfactual distribution F2004|2014
Y in Eq. (3.3) can be
written as F2004|2014,1
Y where 1 = (2014, 2014). In other words, the compu-
tation of the counterfactual distribution F2004|2014
Y uses the conditional dis-
tribution of the outcome given the covariates in year 2004, the dependence structure of year 2014 , and the marginal distributions of the covariates in year 2014. Similarly, we can get F2004
Y = FY2004|2004,0where 0 = (2004, 2004).
Now we can write the composition effect ∆ν X as ∆νX = ν FY2004|2014 − ν FY2004 = ν FY2004|2014,1 − ν FY2004|2004,0 = ν FY2004|2014,1 − ν FY2004|2004,1 + ν FY2004|2004,1 − ν FY2004|2004,0 = ∆ν D+ βν(1) (3.7)
The first term of the decomposition in Eq. (3.7), or ∆ν
D = ν
FY2004|2014,1− νFY2004|2004,1,
captures the contribution of the between-year difference of the covariates’ copula functions. ∆ν
D is thus a dependence effect. The second term, or
βν(1) = νFY2004|2004,1− νFY2004|2004,0
measures the joint contribution of between-year differences in the marginal covariate distributions.
Let now e1 = (2014, 2004) and e2 = (2004, 2014). βν(1) can in turn be
decomposed as βν(1) = βν(1) − βν(e1) − βν(e2)+ βν(e1) + βν(e2) (3.8) with βν(e1) = νFY2004|2004,e1− νFY2004|2004,0 and βν(e2) = ν FY2004|2004,e2− νFY2004|2004,0
In other words, βν(e1) and βν(e2) measure the respective direct contributi-
ons of the first and second covariate. Let ∆ν
M(1) ≡ βν(1) −βν(e
1) −βν(e2).
∆ν
M(1) can then interpreted as a “pure” interaction effect.
To sum up, the composition effect can be written as ∆ν X = βν(e 1 ) + βν(e2 ) + ∆ν M(1) + ∆νD, (3.9)
i.e, as the sum of the respective contributions of each covariate, a term measuring the pure effect of their interaction, and a term measuring the contribution due to the between-year variation of the dependence between covariates. This decomposition can easily be generalized in the case of more than two covariates and focus either on individual effect of each of them and the pure effect of their interaction as shown above, or on the effect of groups of variables and of the interaction among these groups.